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We consider periodic homogenization problems for the L{\'e}vy operators withasymmetric L{\'e}vy densities. The formal asymptotic expansion used for the$\a$-stable (symmetric) L{\'e}vy operators ($\a\in (0,2)$) is not applicabledirectly to such asymmetric cases. We rescale the asymmetric densities, extractthe most singular part of the measures, which average out the microscopicdependences in the homogenization procedures. We give two conditions (A) and(B), which characterize such a class of asymmetric densities, that the above"rescaled" homogenization is available.

Homogenizations of integro-differential equations with Lévy operators with asymmetric and degenerate densities